Microscopic evaluation of spin and orbital moment in ferromagnetic resonance

Time-resolved X-ray magnetic circular dichroism under the effects of ferromagnetic resonance (FMR), known as X-ray ferromagnetic resonance (XFMR) measurements, enables direct detection of precession dynamics of magnetic moment. Here we demonstrated XFMR measurements and Bayesian analyses as a quantitative probe for the precession of spin and orbital magnetic moments under the FMR effect. Magnetization precessions in two different Pt/Ni-Fe thin film samples were directly detected. Furthermore, the ratio of dynamical spin and orbital magnetic moments was evaluated quantitatively by Bayesian analyses for XFMR energy spectra around the Ni \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{2,3}$$\end{document}L2,3 absorption edges. Our study paves the way for a microscopic investigation of the contribution of the orbital magnetic moment to magnetization dynamics.


S1. Magnetic precession in FMR detected by XFMR
Magnetization precession in the ferromagnetic resonance is expressed by the Landau-Lifshitz-Gilbert (LLG) equation, which is given by, where M represents a magnetic moment vector, M is a norm of M, and µ 0 , γ, and α are magnetic permeability of vacuum, electron gyromagnetic ratio, and Gilbert damping factor, respectively.H eff is effective magnetic field including a bias magnetic field H DC , RF magnetic field h AC , and demagnetization field along the surface normal of the sample.Here we defined that H DC is applied along the z direction and the incident X-ray beam is along the y direction as shown in Fig. 1 (b) in the main text.h AC is produced around the signal line of the coplanar waveguide, so that it can be regarded that h AC is approximately along the x direction for the sample.In this geometry, it can be assumed that magnetization precession occurs in the xy plane; that is to say, magnetic moment vector can be defined as M = (m x , m y , M ) where M >> m x , m y .H eff is expressed as, where h x = hx exp(iωt), hx is the amplitude of h AC , ω = 2πf is the angular frequency of the RF field.Demagnetization field is represented by y component of H eff .
Because XMCD measurement provides the information of a component of a magnetic moment along the direction of the incident X-rays, XFMR measurement detects y-components of the M, which is expressed by using the off-diagonal component of the in-plane magnetic susceptibility χ yx as below, According to Eq. (S1.1), real and imaginary parts of χ yx , χ yx 1 and χ yx 2 , can be expressed as below, and where where

S2. Application of XMCD equations for the XFMR measurements
Here we describe an application of XMCD equations for the present XFMR experiments.In the present experiments, microwaves excite magnetization precession continuously, and X-rays with a pulse width of 50 ps detect the moments at each delay time.This time scale is much longer than those of the electronic repopulation and thermalization (δt < 1 ps), which have been investigated by ab-initio calculations for pulse laser pump and x-ray probe experiments [1], as well as inter-band (2p-3d) electronic transition in XMCD process (δt < 1 fs).Thus, our XFMR measurement detects the magnetic moments at an almost steady state at each delay time.XMCD signal (∆γ), which is a difference between X-ray attenuations acquired with left and right circular polarized X-rays, is expressed by the below equations [2]; We assumed that H DC and incident X-ray beams are parallel to z-and y-axes, respectively, as shown in To simplify, we now consider a one-electron process.The spin state of the final state can be given by where θ and ϕ are polar coordinates, shown in Fig. S1.| 1 2 , ↑⟩ z(y) is a spin state of an electron with bases when the z (y) direction is chosen as a quantization axis.Using this notation, |b⟩ can be expressed by where C L,m l ,S,ms is a Clebsch-Gordan coefficient.Defining q ± as then, for our XFMR measurement, F 1 ±1 can be expressed using q ± as below; For the cases that spin is perpendicular to the X-ray direction, i.e., θ = 0, or θ = π/2 and ϕ = 0, Eqs.(S2.11) and (S2.12) lead to 2 , one can calculate as below; where Eq. (S2.15) gives usual XMCD signals when the spin moment points to the y-direction.Thus, according to Eq. (S2.14),XFMR spectra are expressed by projected components of XMCD spectra along the X-ray beam direction.

S3. XMCD Measurements
Figures S2 (a) and (b) present X-ray absorption spectroscopy (XAS) and XMCD data around the Ni L 2,3 edges for the Ta(2)[Pt(2)/Py(5)] 6 and the Pt(10)/Py(30) samples.XAS data were obtained in the totalelectron-yield (TEY) mode.An external magnetic field was applied perpendicular to the sample surface.I + and I − represent the XAS data obtained using right and left circular polarized X-rays, divided by incident (I0) X-ray intensity.XMCD data were acquired by subtracting these spectra.
Figure S3 shows the angular dependence (θ = 0 • , 70 • ) of XMCD data for the Pt(10)/Py(30) sample.θ is defined as an angular between the direction of the incident X-ray and the surface normal of the sample.All spectra are normalized by the intensities at the L 2 edge.The intensity at the L 3 edge is decreased at

Fig. S1 .
In Eqs.(S2.1)∼(S2.3),|a⟩ and |b⟩ represent the initial and the intermediate states with the energies E a and E b , respectively.R±1 is defined as R±1 = ∓1/ √ 2( Rz ± i Rx ) (and R0 = Ry ) with Cartesian components of the position operator Rx,y,z .p(b), E p , Γ, and C refer to probability of the final state, the energy of the photon, the decay width, and coefficient, respectively.

Fig. S3 :
Fig. S3: Angular (θ) dependence of XMCD signals of the Pt(10)/Py(30).The inset shows the definition of θ These spectra are normalized by the intensities at the L 2 edge.Dashed lines present the intensities at the L 3 edge.All data were acquired by the TEY method.

Defining D =
{E i , I XFMR i } as a obtained dataset, where E i and I XFMR i represent incident X-ray energy and XFMR intensity data, and θ as a parameter set, Bayes' theorem between D and θ is expressed by below equation [3, 4, 5]: p(θ|D, b) = p(D|θ, b) p(θ) p(D, b) , (S4.1)